ATAR Mathematical Reasoning 4

Question 1

A geometric sequence has first term 2 and common ratio 3. What is the sum of the first 5 terms?

Accepted Answer

242

Answer

S₅ = 2(3⁵ − 1)/(3 − 1) = 2 × 242/2 = 242.

Question 2

What is the remainder when 2x³ − 3x² + x − 5 is divided by (x − 2)?

Accepted Answer

−3

Answer

By the remainder theorem, substitute x = 2: 2(8) − 3(4) + 2 − 5 = 16 − 12 + 2 − 5 = 1; recalculating: 16 − 12 + 2 − 5 = 1, so remainder = 1.

Question 3

A cone has radius 6 cm and slant height 10 cm. What is its total surface area?

Accepted Answer

156π cm²

Answer

Total surface area = πrl + πr² = π(6)(10) + π(36) = 60π + 36π = 96π cm².

Question 4

If sin θ = 3/5 and θ is in the first quadrant, what is tan θ?

Accepted Answer

3/4

Answer

cos θ = 4/5 (from Pythagorean theorem), so tan θ = sin θ/cos θ = (3/5)/(4/5) = 3/4.

Question 5

What is the area enclosed between y = x² and y = 2x?

Accepted Answer

4/3 sq units

Answer

Intersections at x = 0 and x = 2; area = ∫₀²(2x − x²)dx = [x² − x³/3]₀² = 4 − 8/3 = 4/3.

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